zbMATH — the first resource for mathematics

Towards classification of multiple-end solutions to the Allen-Cahn equation in \(\mathbb{R}^2\). (English) Zbl 1262.35011
Summary: An entire solution of the Allen-Cahn equation \(\Delta u = f(u)\), where \(f\) is an odd function and has exactly three zeros at \(\pm 1\) and \(0\), e.g. \(f(u) = u(u^2 - 1)\), is called a \(2k\)-ended solution if its nodal set is asymptotic to \(2k\) half lines, and if along each of these half lines the function \(u\) looks (up to a multiplication by \(-1\)) like the one dimensional, odd, heteroclinic solution \(H\), of \(H'' = f(H)\). In this paper we present some recent advances in the theory of the multiple-end solutions. We begin with the description of the moduli space of such solutions. Next we move on to study a special class of these solutions with just four ends. A special example is the saddle solutions \(U\) whose nodal lines are precisely the straight lines \(y = \pm x\). We describe the connected components of the moduli space of \(4\)-ended solutions. Finally we establish a uniqueness result which gives a complete classification of these solutions. It says that all \(4\)-ended solutions are continuous deformations of the saddle solution.

35B08 Entire solutions to PDEs
35Q79 PDEs in connection with classical thermodynamics and heat transfer
35J61 Semilinear elliptic equations
PDF BibTeX Cite
Full Text: DOI